Power one sequential tests exist for weakly compact P against P c

We study power-one sequential testing for an i.i.d. law on a Polish sample space. Given a nonempty composite null class M 1, we ask when there exists a level- α stopping rule that rejects almost surely under every alternative in a prescribed class c. Our main sufficient condition is local weak lower semicontinuity and positivity of the information projection functional Φ(Q):= P in (QP). In particular, if is weakly compact, then for every α in(0,1) there is a single level- α sequential test with power one against the entire complement c. The proof combines Csiszár's nonasymptotic Sanov bound for weakly closed convex empirical-measure sets with a Lindelöf countable-subcover argument. We also show that weak lower semicontinuity is sufficient but not necessary by giving examples where discontinuous finite-sample events separate alternatives that weak neighborhoods cannot detect. Finally, we construct an e-process that is asymptotically relatively growth-rate optimal under weak compactness. We verify the weak-lower-semicontinuity condition for weakly compact nulls, f-divergence balls, several integral probability metric balls, Wasserstein balls on proper spaces, and a number of non-weakly-compact semiparametric examples.